LatticeModels.jl

This package provides a set of tools to simulate different quantum lattice systems.

Installation

pkg> add https://github.com/aryavorskiy/LatticeModels.jl

import Pkg; Pkg.install(url="https://github.com/aryavorskiy/LatticeModels.jl")

Package features

Bravais lattices with arbitrary geometry and any possible count of internal states on one sites.
Versatile hamiltonian generation tools.
Backend-independent computations: linear operators can be of any array type, allowing to use sparse or GPU arrays when needed.
Smart unitary evolution macro reducing excessive computations where possible.
Plots.jl integration.

Usage examples

Local density for lowest states in a tight-binding model

The tight-binding model hamiltonian is defined by this formula:

\[\hat{H} = \sum_\text{x-bonds} c^\dagger_i c_j + \sum_\text{y-bonds} c^\dagger_i c_j + h. c.\]

Here we will find its eigenstates and plot their local density on heatmaps.

using LatticeModels
using LinearAlgebra, Plots
# Generate a 40x40 square lattice
l = SquareLattice(40, 40)
# Define the tight-binding model hamiltonian
H = @hamiltonian begin
    lattice := l
    # Add hoppings along axis x and y
    @hop axis = 1
    @hop axis = 2
end

# Calculate eigenvalues and eigenvectors
sp = spectrum(H)

n = 5
clims = (0, 0.0045)
p = plot(layout = @layout[ grid(n, n) a{0.1w}], size=(1000, 850))
for i in 1:n^2
    E_rounded = round(eigvals(sp)[i], sigdigits=4)
    plot!(p[i], site_density(sp[i]), title="\$E_{$i} = $E_rounded\$", clims=clims, cbar=:none)
end

# The following 2 lines are kinda hacky; they draw one colorbar for all heatmaps
plot!(p[n^2+1], framestyle=:none)
scatter!([NaN], zcolor=[NaN], clims=clims, leg=:none, cbar=:right, background_subplot=:transparent,
    framestyle=:none, inset=bbox(0.0, 0.05, 0.95, 0.9), subplot=n^2+2)

Currents in a tight-binding model on a ring-shaped sample

In this example we delete part of the sites in the middle of a square lattice. Then we adiabatically turn on magnetic field through the hole and see currents emerge.

The tight-binding hamiltonian is the same as in the example above.

using LatticeModels
using LinearAlgebra, Plots

l = SquareLattice(10, 10) do site, (x, y)
    abs(x) > 1 || abs(y) > 1
end
h(B) = @hamiltonian begin
    lattice := l
    @hop axis = 1
    @hop axis = 2
    # Add magnetic field through (0, 0) point
    field := FluxField(B, (0, 0))
end
sp = spectrum(h(0))

# Find density matrix for filled bands (e. g. energy < 0)
P_0 = filled_projector(sp)
# Perform unitary evolution
τ = 10
a = Animation()
@evolution {
    H := h(0.1 * min(t, τ) / τ)
    P_0 --> H --> P
} for t in 0:0.1:2τ
    # Find the partial trace and plot it
    plot(site_density(P), clims=(0,1))

    # Show currents on the plot
    plot!(DensityCurrents(H, P), arrows_scale=7, arrows_rtol=0.1)

    title!("t = $t")
    frame(a)
end

gif(a, "animation.gif")

Local Chern marker with hamiltonian quench

The Chern insulator hamiltonian is described by this formula:

\[\hat{H} = \sum_i m_i c^\dagger_i \sigma_z c_i + \left( \sum_\text{x-bonds} c^\dagger_i \frac{\sigma_z - i \sigma_x}{2} c_j + \sum_\text{y-bonds} c^\dagger_i \frac{\sigma_z - i \sigma_y}{2} c_j + h. c. \right)\]

In this experiment we create a filled state density matrix for a system with $m_i = 1$. After that we change the $m_i$ in the center of the lattice to $-1$ and start the evolution.

using LatticeModels
using LinearAlgebra, Plots

l = SquareLattice(11, 11)
x, y = coord_values(l)

# The Pauli matrices
σ = [[0 1; 1 0], [0 -im; im 0], [1 0; 0 -1]]

# Initial hamiltonian: m=1 everywhere
H1 = @hamiltonian begin
    lattice := l
    dims_internal := 2
    @diag σ[3]
    @hop (σ[3] - im * σ[1]) / 2 axis = 1
    @hop (σ[3] - im * σ[2]) / 2 axis = 2
end

# Quenched hamiltonian: m=-1 in the central 3x3 square
M = @. (abs(x) < 1.5 && abs(y) < 1.5) * -2 + 1
H2 = @hamiltonian begin
    lattice := l
    dims_internal := 2
    @diag M ⊗ σ[3]
    @hop (σ[3] - im * σ[1]) / 2 axis = 1
    @hop (σ[3] - im * σ[2]) / 2 axis = 2
end
X, Y = coord_operators(l, 2)

sp = spectrum(H1)
P_0 = filled_projector(sp)

τ = 10
a = Animation()
@evolution {
    H := H2
    P_0 --> H --> P
} for t in 0:0.1:2τ
    p = plot(layout=2, size=(900, 400))

    # Local Chern marker heatmap
    lcm_operator = 4pi * im * P * X * P * Y * P
    chern_marker = ptrace(lcm_operator) .|> real
    plot!(p[1], chern_marker, clims=(-2, 2))

    # Select sites on y=0 line (use ≈ to avoid rounding errors)
    chern_marker_on_sw = chern_marker[@. y ≈ 0]
    # Mark selected sites on the heatmap
    plot!(p[1], lattice(chern_marker_on_sw), high_contrast=true)
    # Add a line plot
    plot!(p[2], project(chern_marker_on_sw, :x), ylims=(-3, 3), lab=:none)

    plot!(plot_title="t = $t")
    frame(a)
end

gif(a, "animation.gif")

LDOS animation

Local density can be a bit ambiguous for degenerate eigenstates. That's where the LDOS (Refer to ldos documentation) will be helpful.

Let's take the same hamiltonian from the previous example and create a LDOS animation.

using LatticeModels
using LinearAlgebra, Plots
l = SquareLattice(40, 40)
σ = [[0 1; 1 0], [0 -im; im 0], [1 0; 0 -1]]
H = @hamiltonian begin
    lattice := l
    dims_internal := 2
    @diag σ[3]
    @hop (σ[3] - im * σ[1]) / 2 axis = 1
    @hop (σ[3] - im * σ[2]) / 2 axis = 2
end

sp = spectrum(H)
δ = 0.1
Es = -4:0.1:4
Es_d = -4:0.01:4
ldosf = ldos(sp, δ)
a = @animate for E in Es
    p = plot(layout=2, size=(800, 400))
    plot!(p[1], Es_d, dos(sp, δ), lab="", title="DOS")
    vline!(p[1], [E], lab="")
    plot!(p[2], ldosf(E), clims=(0, NaN), title="LDOS")
    plot!(p, plot_title="E = $E, δ = $δ")
end

gif(a, "animation.gif", fps=10)